Crushing one theorem at a time

The Exponential and Trigonometric Functions

Point of Post: In this post we define the exponential and trigonometric functions and note that they are holomorphic.

Motivation

Last time we proved that every function on an open subset of that is locally representable by power series is necessarily holmorphic (in fact, infinitely differentiable!). That said, we didn’t actually give any honest to god examples of such functions. Thus, in this post we will finally lay down our first few non-trivial examples of holomorphic functions. They will come in terms of what is, in a very precise sense we will make clear later on, the only extension of some of our favorite real valued functions: the exponential and trigonometric functions.

The Exponential and Trigonometric Functions

Let us define the exponential function by the rule

We shall often denote as . Note that since

that the radius of convergence of is and so it really is a well-defined function on .

I leave it to the reader to verify that is a homomorphism , or more explicitly that and so in particular for all since . Moreover, I leave it for you to check that (just use the fact that the conjugation map is continuous) and that for integers one has that for all .

Since is representable by power series everywhere on we know that –such globally holomorphic functions are called entire. We note moreover that, by definition,

so that . In fact, we have the following theorem:

Theorem: The exponential function is the unique entire function such that and .

Proof: It clearly suffices to show uniqueness. Indeed, if were such a function then we’d have that is entire (note we can divide since is never zero). That said, using the division rule we see that

since and . Thus, since is connected we may conclude from previous proof that is constant. Noting that we may conclude that as desired.

What we would now like to do is explain how the exponential and trigonometric functions are related. We begin with the observation that if is real then

and similarly

And thus, in particular,

References:

[1] Greene, Robert Everist, and Steven George Krantz. Function Theory of One Complex Variable. Providence: American Mathematical Society, 2006. Print.